Inner Product Spaces

Abstract

So far we have seen that the definition of an abstract vector space captures the fundamental geometric notion of dimension. There remain, however, at least two other basic geometric ideas that we have not yet addressed: length and angle. To encompass them in the abstract we need to introduce a bit more structure, and in consequence we shall require that our ground field manifest some notion of order. Hence we no longer operate over some abstract field k but rather, for the most part, over the field R of real numbers. In the final section we shall generalize the results to the complex numbers C.